Fact-checked by Grok 2 weeks ago

Domain wall

A domain wall is a type of topological defect, specifically a (D-1)-dimensional interface separating regions of space with different vacuum states that arises whenever a discrete symmetry is spontaneously broken in field theories.^[1] In condensed matter physics, it manifests as a thin interfacial region separating adjacent domains in ordered materials, such as ferromagnets or ferroelectrics, where the material's order parameter—such as magnetization or electric polarization—rotates or varies continuously to transition between distinct uniform states, minimizing overall energy through a balance of exchange, anisotropy, and electrostatic contributions.^[2] These walls typically span nanometers to tens of nanometers in thickness and exhibit topological properties, ensuring that any path connecting points across the wall must traverse the transitional region. In ferromagnetic materials, domain walls form to reduce demagnetizing fields, with common types including Bloch walls, where magnetization rotates in the plane perpendicular to the wall, and Néel walls, featuring in-plane rotation suitable for thin films to avoid stray fields.^[2] Similarly, in ferroelectrics, walls separate regions of differing polarization directions, categorized as 180° (reversing polarization) or non-180° (e.g., 90° or 109° in materials like BaTiO₃ or BiFeO₃), with charged variants carrying bound charge at head-to-head or tail-to-tail configurations that can enable enhanced conductivity.^[3] The energy density of these walls, often on the order of 10^{-3} J/m² in iron, governs domain sizes via relations like Kittel's law, where domain width scales with the square root of film thickness.^[2] Domain walls play a pivotal role in the macroscopic properties of ferroic materials, influencing hysteresis, coercivity, and response to external fields, while defects or strains can pin walls, affecting stability. Beyond fundamental physics in materials, they also appear in cosmology as relics from early universe phase transitions with potential gravitational effects, and in theoretical physics contexts like string theory branes or domain wall fermions in lattice QCD.^[4] They enable advanced applications in nanoelectronics, such as racetrack memory devices exploiting wall motion for data storage in ferromagnets, and conductive nanochannels in ferroelectrics for resistive switching with on/off ratios exceeding 10^5.^[3] In multiferroics like BiFeO₃, coupled magnetic and electric properties at walls further promise hybrid devices for spintronics and sensors.

General Concept

Definition

A domain wall is a type of topological soliton in field theory that forms a (d-1)-dimensional boundary separating spatial regions, or domains, where the order parameter or vacuum state differs between two degenerate ground states.^[5] These structures arise as stable configurations in systems where the field's potential has multiple isolated minima, allowing the field to transition smoothly from one vacuum to another across the wall.^[6] Domain walls occur whenever a discrete symmetry, such as Z₂, is spontaneously broken, resulting in degenerate vacua that the system selects locally during the symmetry-breaking process.^[6] In the simplest case, like the φ⁴ theory with a double-well potential invariant under φ → -φ, the Z₂ symmetry breaking leads to two equivalent vacua at φ = ±v, and a domain wall interpolates between them.^[5] In 1+1 dimensions, domain walls manifest as kinks—localized, particle-like excitations—as exemplified by the sine-Gordon model, where the field winds between vacua related by a discrete shift. Extending to higher dimensions, they become extended sheet-like defects spanning two spatial directions while localizing in the third, preserving their topological character.^[5] These solitons exhibit localized energy density concentrated near the interface, with topological stability ensuring they cannot decay into the uniform vacuum without violating conservation laws, provided the symmetry breaking is exact.^[6] In scenarios with approximate breaking, such as when a small explicit symmetry-violating term introduces false vacua, domain walls may become metastable or unstable, potentially annihilating over time.^[5]

Formation and Stability

Domain walls form during spontaneous symmetry breaking phase transitions when a system is driven through a critical point, such as via a quench in temperature or coupling strength. In this process, the Kibble-Zurek mechanism governs the dynamics: causal horizons limit the correlation length, leading to independent selection of the broken symmetry vacuum in spatially separated regions, with domain walls emerging as topological boundaries between these randomly chosen domains.^[7] The density and size of domains, and thus the wall network, depend critically on the quench rate. Slower cooling allows larger correlation lengths to develop before the transition, reducing wall density, while rapid quenches produce smaller domains and more walls. The characteristic domain size scales as \xi \sim \tau_Q^{\nu / (1 + \nu z)}, where \tau_Q is the quench time, and \nu and z are the critical exponents for the correlation length and dynamics, respectively.^[8] Domain walls are topologically stable when the underlying discrete symmetry is exact, as their existence is protected by the non-trivial homotopy group \pi_0 of the vacuum manifold, preventing continuous deformation to the uniform vacuum without energy cost. However, if the symmetry is approximate—due to explicit breaking terms—the walls become metastable and can decay. For instance, a small bias \epsilon in the scalar potential V(\phi) = \frac{\lambda}{4} (\phi^2 - v^2)^2 + \epsilon \phi tilts the degenerate minima, introducing a pressure difference across the wall that drives its motion and eventual annihilation.^[7] In cases of approximate symmetry, unstable domain walls accelerate under the bias, collide, and annihilate, releasing their energy primarily as scalar radiation or, in particle physics contexts, into Standard Model particles; this contrasts with exact symmetry scenarios, where walls persist indefinitely, potentially leading to cosmological overclosure. Decay can also occur via quantum tunneling through the energy barrier for sufficiently thin walls or thermal activation at finite temperatures.^[9]^[10] This formation mechanism applies broadly, including to phase transitions in the early universe and to magnetic domain formation in materials cooled from the paramagnetic state.^[11]

Magnetic Domain Walls

Structure and Types

In ferromagnetic materials, a magnetic domain wall forms a narrow transition region, typically spanning about 100-150 atoms thick, across which the magnetization vector rotates continuously from the direction in one domain to that in an adjacent domain, often involving angular displacements of 90° or 180°.^[12] This rotation occurs gradually to minimize the total magnetic energy, creating a coherent spin structure that connects uniformly magnetized regions. The internal profile of the wall is influenced by the balance of exchange and anisotropy energies, which dictate the smoothness of the spin reorientation.^[13] Domain walls are classified primarily by their geometric configurations, with Bloch and Néel walls being the two canonical types in ferromagnets. In a Bloch wall, the magnetization rotates within the plane of the wall (perpendicular to the wall normal), resulting in a transverse component that lies within the wall plane and perpendicular to both the domain magnetization and the wall normal; this configuration effectively reduces demagnetization energy by canceling stray fields and is prevalent in bulk materials or thick samples where volume effects dominate.^[14]^[12] Originally proposed by Felix Bloch, this type accommodates the rotation without generating significant long-range dipolar fields in three-dimensional structures.^[15] In contrast, a Néel wall features magnetization rotation within a plane containing the wall normal (perpendicular to the wall plane), producing a component parallel to the wall normal; this arrangement is favored in thin films or nanowires to suppress stray magnetic fields that would otherwise arise from out-of-plane components in low-dimensional geometries.^[16]^[13] Named after Louis Néel, who analyzed domain structures in thin magnetic layers, Néel walls become energetically preferable when the sample thickness is comparable to or smaller than the wall width, as the rotation aligns spins to minimize demagnetizing contributions from perpendicular components.^[16] Within these walls, finer substructures known as domain lines serve as topological defects that further modulate the spin configuration. Néel lines appear as localized transitions between segments of Bloch walls, where the transverse magnetization component reverses direction, while Bloch lines mark chirality changes in Néel walls, introducing twists in the in-plane rotation.^[17] These lines act as stable point-like defects embedded in the extended wall, influencing local spin textures and contributing to the overall topological stability of the structure.^[18] The microscopic structure of domain walls can be visualized using techniques that probe magnetic stray fields or phase shifts in electron waves. The Bitter method involves applying a colloidal suspension of ferromagnetic particles to the sample surface, where the particles align along field lines to outline wall positions and contours under optical microscopy.^[19] For higher resolution, electron holography reconstructs the magnetic induction map by interfering an electron beam with a reference wave, revealing the vectorial spin distribution and defect lines within the wall with atomic-scale precision.^[20] These methods have confirmed the predicted geometric features in materials like iron and permalloy. Domain walls play a key role in magnetization reversal by nucleating and propagating to switch domain orientations under applied fields.^[13]

Energy and Width

The surface energy density σ of a magnetic domain wall quantifies the excess energy per unit area associated with the transition between adjacent magnetic domains, arising from the competition between exchange interactions, which favor parallel spin alignment and thus a broader transition region to minimize spatial gradients, and magnetocrystalline anisotropy, which favors spin alignment along preferred crystallographic directions and thus a sharper transition to reduce angular deviations. This balance determines the equilibrium structure of the wall, as originally derived by Bloch in the 1930s.^[21] For a 180° wall in uniaxial materials, the minimized total energy yields σ = 4 √(A K), where A is the exchange stiffness constant and K is the uniaxial anisotropy constant.^[22] The exchange energy contribution is expressed in the continuum limit as E_ex = A ∫ (∇m)^2 dV over the volume, where m is the unit vector of magnetization direction; this term increases with steeper spin gradients. In a discrete Heisenberg model for a one-dimensional chain of N spins spanning the wall, the exchange energy approximates to E_ex ≈ (J_ex S^2 π^2)/N, where J_ex is the nearest-neighbor exchange integral and S is the atomic spin magnitude, highlighting how finer discretization (larger N) reduces the energy cost.^[23] The competing anisotropy energy is given by E_an = K ∫ sin^2 θ dV, where θ is the angle between the magnetization and the easy axis; this term penalizes rotations away from the easy directions within the wall. Minimizing the sum of these energies with respect to the wall profile leads to the characteristic wall width δ ≈ π √(A / K), which typically falls in the range of 10–100 nm for common ferromagnetic materials like iron or permalloy, depending on their microscopic parameters. The equilibrium magnetization profile across the wall, assuming a one-dimensional variation along z perpendicular to the wall plane, follows θ(z) = 2 \arctan\left( \exp\left( z / \Delta \right) \right), where Δ = √(A / K) sets the intrinsic length scale of the transition. This width and the associated energy can be modulated by material-specific factors, including the saturation magnetization M_s, which contributes to magnetostatic (demagnetization) energy that favors certain wall configurations, and external magnetic fields, which alter the effective anisotropy landscape. For instance, Bloch and Néel walls exhibit differing total energies primarily due to variations in their magnetostatic contributions.^[22]

Dynamics and Applications

The dynamics of magnetic domain walls under external influences, such as applied magnetic fields, reveal distinct regimes of motion governed by the interplay of gyromagnetic precession, damping, and material parameters. In the Walker model, which describes the steady-state motion of a 180° Bloch wall in a uniaxial ferromagnet under a uniform dc magnetic field H applied along the easy axis, the wall velocity v below the Walker breakdown field H_W is given by v = \frac{\gamma \Delta}{\alpha} H, where \gamma is the gyromagnetic ratio, \Delta is the wall width, and \alpha is the Gilbert damping parameter, with H_W = \frac{\alpha}{2} H_a and H_a the anisotropy field. Above H_W, the wall undergoes oscillatory precession, leading to a maximum velocity v_W \approx \frac{\gamma \Delta H_W}{\alpha} and chaotic dynamics, limiting steady propagation. In real materials, defects and inhomogeneities introduce pinning effects that trap domain walls, requiring a critical depinning field H_p to initiate motion. Below H_p, walls exhibit thermally activated creep motion, where velocity follows an exponential dependence v \propto \exp\left(-\frac{U_c}{k_B T}\right) with U_c the pinning energy barrier, characteristic of glassy, logarithmic dynamics over long times.^[24] At H > H_p, walls enter a sliding or flow regime with linear velocity-field response, transitioning to steady propagation dominated by viscous damping rather than disorder.^[24] These regimes have been observed in thin films via magneto-optical imaging, highlighting the role of pinning landscapes in controlling wall mobility.^[25] Applications of domain wall dynamics leverage controlled motion for beyond-CMOS computing and storage. Domain wall logic devices use intersecting nanowires where wall propagation encodes Boolean operations, such as AND/OR gates, by exploiting pinning sites and field-driven shifts for low-power, non-volatile processing. In racetrack memory, proposed by IBM in the 2000s, data bits are stored as domain wall positions along ferromagnetic nanowires; spin-transfer torque from current pulses shifts walls at speeds up to 100 m/s, enabling dense, 3D-integrated storage with read/write via magnetic tunnel junctions. Recent advances focus on electrical manipulation to reduce power consumption and enable scalability. Spin-orbit torque (SOT) in heavy metal/ferromagnet bilayers, such as Pt/Co, generates efficient damping-like and field-like torques from spin Hall effects, driving chiral Néel walls at velocities exceeding 1 km/s without external fields, surpassing spin-transfer torque limits.^[26] Additionally, multiferroic domain walls exhibit coupled magnetic and electric properties, predicted through analysis of the 64 magnetic point groups allowing spontaneous polarization or magnetization at walls, enabling voltage control of magnetism in materials like BiFeO₃ for hybrid spintronic devices. As of 2025, dual spin-orbit torques in synthetic antiferromagnets have been demonstrated to efficiently drive domain wall motion, enhancing control in spintronic devices.^[27]

Cosmological Domain Walls

Origins in Particle Physics

Cosmological domain walls originate from the spontaneous breaking of discrete symmetries in high-energy particle physics models, such as grand unified theories (GUTs) and their supersymmetric extensions, occurring during phase transitions in the early universe at energy scales around $10^{15} GeV.90033-X) These defects were first proposed by Tom Kibble in 1976 as part of a broader class of topological defects—alongside monopoles and cosmic strings—that arise from the Kibble mechanism during such transitions, where correlated symmetry breaking in causally disconnected regions leads to differing vacua separated by walls. In GUTs like SU(5) or SO(10), discrete symmetries such as Z_N (where N is an integer) emerge from the structure of the Higgs sector or fermion representations, and their breaking produces thin sheet-like domain walls with surface tension \sigma \sim \eta^3, where \eta is the vacuum expectation value associated with the symmetry scale.91322-H) Supersymmetric GUTs amplify this possibility, as extended symmetries like Z_2^R or higher Z_N variants can lead to stable vacua configurations that favor wall formation unless mitigated.91322-H) Specific examples illustrate this origin in beyond-Standard-Model physics. In QCD axion models addressing the strong CP problem, the Peccei-Quinn symmetry is anomalous, effectively restoring a discrete Z_N symmetry (with N related to the domain wall number, often N \geq 6 for color-triplet quarks), whose spontaneous breaking at the axion scale (\sim 10^{12} GeV) generates multiple degenerate vacua separated by domain walls. Similarly, in flavor physics models employing non-Abelian discrete groups like A_4 (the alternating group on four elements) to explain lepton mixing patterns, the breaking of this symmetry at the electroweak scale (\sim 100 GeV) during the post-electroweak phase transition can produce domain walls, particularly in supersymmetric extensions where flavons acquire vacuum expectation values that lift degeneracies. These walls interpolate between distinct vacua, such as those differing by permutations in the lepton sector, and their formation is tied to the group's tetrahedral structure. In the post-inflationary era, if the symmetry-breaking transition occurs after the end of inflation—typically during reheating when the universe is radiation-dominated—domain walls form with initial separations on the order of the Hubble scale H^{-1}, reflecting the causal horizon at the transition epoch.90033-X) To prevent cosmological overdominance by these walls, models often incorporate a small explicit breaking of the discrete symmetry, such as through Planck-suppressed higher-dimensional operators (e.g., dimension-5 terms involving the scalar field divided by M_{\rm Pl}), which introduces a bias that favors one vacuum and destabilizes the walls over time.30792-7) This bias mechanism, first highlighted in the context of axion cosmology, ensures the walls annihilate before Big Bang nucleosynthesis without leaving excessive relics. Alexander Vilenkin's 1985 review further elaborated on these dynamics, emphasizing how such biases resolve stability issues in GUT-inspired scenarios.90033-X)

Cosmological Consequences

In cosmology, stable domain walls pose a significant challenge due to their energy density scaling. The energy density of a domain wall network in the scaling regime is given by \rho_\text{wall} \sim \sigma / d, where \sigma is the wall tension and d is the typical inter-wall distance, which evolves proportionally to the horizon size, leading to \rho_\text{wall} \propto t^{-1} or \rho_\text{wall} \sim \zeta \sigma H with \zeta \approx 1-10.^[28]^[29] This slower dilution compared to radiation (\rho_r \propto t^{-2}) or matter (\rho_m \propto t^{-3/2}) causes walls to dominate the energy budget if they persist, inducing an accelerated expansion phase with a(t) \propto t that conflicts with the observed flat, matter-radiation dominated universe and disrupts Big Bang nucleosynthesis.^[30]^[31] This constitutes the "domain wall problem," first highlighted in the context of grand unified theories where such defects could form copiously. To resolve this issue, several mechanisms have been proposed. Inflation prior to the symmetry-breaking transition can exponentially dilute the wall density, suppressing their abundance below dangerous levels if the reheating temperature is below the wall formation scale.^[32] For transitions occurring after inflation, such as in axion models, a small bias term \epsilon in the scalar potential explicitly breaks the discrete symmetry, introducing a pressure difference between vacua that drives biased annihilation of the wall network.^[33] This process releases the stored energy primarily into gravitational waves (GWs) and potentially relativistic particles or non-thermally produced dark matter candidates like axions from wall collapse. Alternatively, hybrid topological defects—domain walls bounded by cosmic strings—can form in certain symmetry-breaking chains, such as SO(10) grand unified models; these structures annihilate more rapidly without dominating the energy density, serving as cosmologically harmless configurations.^[34] Observationally, the absence of domain wall signatures imposes stringent constraints. The lack of detected spectral distortions in the cosmic microwave background (CMB) from energy injection by pre-recombination walls limits the tension to \sigma < 10^{15} \, \text{GeV}^3, as higher values would produce \mu-type distortions exceeding current upper bounds from COBE/FIRAS.^[35] Annihilating walls could generate a stochastic GW background peaking in the mHz range, potentially detectable by future space-based interferometers like LISA, with peak amplitudes h^2 \Omega_\text{GW} \sim 10^{-12} - 10^{-10} for \sigma \sim 10^{12} - 10^{15} \, \text{GeV}^3 and annihilation around the electroweak scale.^[9] No such signals have been observed in pulsar timing arrays or ground-based detectors, further tightening bounds on late-time wall networks.^[36] Recent advancements explore quantum gravity effects as a natural bias source. A 2025 study proposes that non-perturbative quantum gravity corrections, such as those from string theory or loop quantum gravity, could induce a tiny explicit breaking of discrete symmetries protecting flavor domain walls, triggering their annihilation and yielding signatures in dark radiation, primordial black holes, and GWs without fine-tuning.^[37] These mechanisms align with high-scale models while evading the domain wall problem through fundamental physics origins.

Domain Walls in Theoretical Physics

In String Theory

In string theory, domain walls manifest as codimension-one brane-like defects that interpolate between distinct vacua in the theory's landscape. These structures arise as singularities separating regions with different asymptotic values of scalar fields or fluxes, exemplified by D8-branes in type IIA string theory, which act as charged domain walls sourcing Roman mass parameters and connecting regions of varying dilaton and Romans mass. Similarly, NS5-branes in type IIB can embed as domain walls, while more general (p,q)-walls represent bound states or T-dual configurations that preserve specific supersymmetries while bridging vacua with differing moduli. These walls are inherently topological, deriving stability from the breaking of discrete symmetries in the underlying field theory, though in string-theoretic embeddings, their persistence is tied to the global structure of the compactification. The dynamics of these domain walls are often described by BPS-saturated solutions that preserve a fraction of the ambient supersymmetry, ensuring stability against small perturbations. In type II theories, half-BPS domain walls in toroidally compactified backgrounds satisfy first-order equations derived from the supergravity action, leading to warped geometries where the wall tension balances gravitational backreaction. An effective low-energy description emerges by viewing the wall as a thick brane sourced by scalar fields in an extra dimension, akin to Randall-Sundrum models, where the scalar potential generates a kink profile that localizes gravity and other zero modes on the wall. Such constructions embed naturally in string theory via flux compactifications, with the wall's thickness regulated by the string scale. Applications of domain walls in string theory model building include their role in moduli stabilization, where sequences of walls connect different flux vacua, fixing otherwise runaway moduli through tension-induced potentials. Intersections of domain walls can produce lower-dimensional defects, such as domain wall strings—effective 1D objects in reduced dimensions arising from the crossing of codimension-one branes, which carry tension and exhibit dynamics analogous to fundamental strings. In scenarios involving axionic fields from string compactifications, the collapse of domain wall networks intertwined with axion strings can generate axion dark matter relics or stochastic gravitational wave backgrounds, providing testable cosmological signatures. Historically, domain walls were introduced in heterotic string models during the 1990s, particularly in five-dimensional gauged supergravity duals of strongly coupled heterotic strings on Calabi-Yau orbifolds, where BPS walls resolved singularities and stabilized the Hořava-Witten setup. More recent work has explored the dynamics of domain wall solitons in 2+1-dimensional effective theories derived from string compactifications, revealing string-like behaviors and interactions that inform higher-dimensional embeddings. These developments link to braneworld cosmology, where domain walls serve as our universe's location in warped extra dimensions.

Domain Wall Fermions

Domain wall fermions represent a lattice discretization method in quantum field theory designed to simulate chiral fermions without the fermion doubling problem inherent in naive lattice formulations. Introduced by David B. Kaplan in 1992, this approach embeds four-dimensional spacetime into a five-dimensional bulk, where an extra fifth dimension x_5 is introduced with a position-dependent mass term m(x_5) = M \operatorname{sign}(x_5), with M > 0. This "domain wall" mass profile localizes massless chiral fermion modes on the wall at x_5 = 0, while massive modes remain gapped in the bulk away from the wall, effectively yielding a four-dimensional theory of chiral fermions confined to the defect.^[38] The method circumvents fermion doubling by transforming the 16 doubler modes of a four-dimensional naive Dirac fermion into heavy Kaluza-Klein excitations in the fifth dimension, which decouple from low-energy physics as their masses become large. Chiral symmetry is approximately preserved, with violations suppressed exponentially in the size L_s of the extra dimension: the residual fermion mass scales as m_{\text{res}} \sim e^{-L_s / \xi}, where \xi is the localization length determined by the domain wall profile. In the limit L_s \to \infty, exact chiral symmetry emerges, providing topological protection for zero modes bound to the wall. This formulation has been implemented in lattice quantum chromodynamics (QCD) simulations, notably for heavy quarks; for instance, a 2017 study explored domain wall fermions for quarks approaching the bottom quark mass regime, demonstrating feasibility on fine lattices with spacings down to 0.044 fm.^[39]^[38] The continuum limit of domain wall fermions corresponds to overlap fermions, which satisfy the Ginsparg-Wilson relation for exact chiral symmetry on the lattice without requiring an infinite extra dimension. Compared to Wilson fermions, which explicitly break chiral symmetry through a Wilson term, or staggered fermions, which suffer from taste-breaking artifacts leading to incomplete chiral symmetry restoration, domain wall fermions offer superior chiral properties essential for accurately simulating weak interactions and anomalies in the Standard Model. However, the computational overhead from the extra dimension—requiring propagation along L_s sites—poses a significant challenge, often making simulations more resource-intensive despite optimizations like the Möbius domain wall variant.^[40]

References

[1]
Domain Wall - an overview | ScienceDirect Topics
A domain wall is defined as a boundary that exists between two regions of differing magnetization, where all paths connecting these regions must pass ...
[2]
Physics and applications of charged domain walls - Nature
Nov 30, 2018 · The charged domain wall is an ultrathin (typically nanosized) interface between two domains; it carries bound charge owing to a change of normal component of ...<|control11|><|separator|>
[3]
[PDF] TASI Lectures on Solitons - DAMTP
Domain walls, also known as kinks, are co-dimension 1 solitons that interpolate between distinct vacua, and are related to monopoles.
[4]
Cosmic strings and domain walls - ScienceDirect
Cosmic strings and domain walls. Author links open overlay panelAlexander ... Vilenkin, A.E. Everett. Phys. Rev. Lett., 48 (1982), p. 1867. View in Scopus.
[5]
Topology of cosmic domains and strings - IOPscience
It is shown that the formation of domain wall, strings or monopoles depends on the homotopy groups of the manifold of degenerate vacua.
[6]
Kibble–Zurek mechanism in colloidal monolayers - PNAS
The Kibble–Zurek mechanism describes the evolution of topological defect structures like domain walls, strings, and monopoles when a system is driven ...
[7]
Gravitational waves from domain walls and their implications
Jul 10, 2017 · In this paper we focus on the annihilation of unstable domain walls, and evaluate its impact on the currently ongoing and planned GW experiments.
[8]
[PDF] Domain Wall as Cosmological Oscillator - arXiv
Apr 30, 2025 · Abstract: In this study, we examine the domain wall within the framework of a cosmological harmonic oscillator.
[9]
Simulating the Kibble-Zurek mechanism of the Ising model ... - Nature
Mar 8, 2016 · The Kibble-Zurek mechanism (KZM) predicts the density of topological defects produced in the dynamical processes of phase transitions in ...
[10]
Structure and Energy of One‐Dimensional Domain Walls in ...
It is found that only Bloch and Néel types of one‐dimensional walls result. The magnetization distribution of a typical Bloch wall is shown, and the Bloch wall ...
[11]
Domain wall nanoelectronics | Rev. Mod. Phys.
Feb 3, 2012 · The domain wall polarization is caused by the spin spiral inherent to the Néel wall. From Logginov et al., 2008 . D. Domain wall roughness ...Missing: Louis | Show results with:Louis
[12]
Zur Theorie des Ferromagnetismus - Semantic Scholar
1,076 Citations · Allgemeine Theorie des Ferromagnetismus · Über die Magnetonenzahlen ferromagnetischer Stoffe · Über die temperaturabhängigkeit der magnetisierung ...Missing: 1932 pdf
[13]
[PDF] FELIX BLOCH - National Academy of Sciences
Bloch worked out the thickness and structure of the boundary walls, and the wall structure became known as the "Bloch Wall." In a space of a few hundred ...
[14]
Louis Néel: His multifaceted seminal work in magnetism
This new type of domain wall, replacing Bloch walls in 2D magnetic films, was called a Néel wall [61]. With regard to the magnetic domains, Néel first tried ...
[15]
Domain Walls in Thin Ni-Fe Films | IBM Journals & Magazine
In thinner films, the domain walls are of the Néel type. The position of Bloch lines in such walls is indicated by crosswalls.
[16]
Bloch-line generation in cross-tie walls by fast magnetic-field pulses
Apr 21, 2006 · They consist of a sequence of circular and cross Bloch lines (also called vortex and antivortex) that are connected by 90° Néel walls.
[17]
[PDF] Techniques to Measure Magnetic Domain Structures
Jun 16, 1999 · By selecting the appropriate Bitter solution, these commercial ferrofluids can reveal magnetic structures down to the resolution limit of the.
[18]
Electron Holography for Advanced Characterization of Permanent ...
Electron holography is an indispensable tool for obtaining information on magnetic flux density and visualizing magnetic domain structures in permanent magnets ...
[19]
Zur Theorie des Austauschproblems und der ...
Cite this article. Bloch, F. Zur Theorie des Austauschproblems und der Remanenzerscheinung der Ferromagnetika. Z. Physik 74, 295–335 (1932). https://doi.org ...
[20]
Magnetic domain walls and the relaxation method - ScienceDirect.com
Domain wall energy is given by 4 AK illustrating once again the competing role of exchange and anisotropy. For bulk materials, walls of the Bloch type occur ...
[21]
[PDF] Magnetic domains
... thickness, i.e., the thickness of domain wall is: δ=Na=√J S. 2 π. 2. K a. ⇒ δ ∝√J. K. ○. For iron the above expressions predict domain wall thickness of approx.
[22]
Domain Wall Relaxation, Creep, Sliding, and Switching in ...
Sep 9, 2002 · dynamical phase transitions, which link the relaxation, creep, sliding, and switching regimes of pinned domain walls. Figure 1; Figure 2; Figure ...
[23]
Pinning of domain walls in thin ferromagnetic films | Phys. Rev. B
Aug 7, 2018 · We present a quantitative investigation of magnetic domain-wall pinning in thin magnets with perpendicular anisotropy.
[24]
Current-induced spin-orbit torques in ferromagnetic and ...
Sep 9, 2019 · This paper reviews recent progress in the field of spin orbitronics, focusing on theoretical models, material properties, and experimental resultsArticle Text · Theory of Spin-orbit Torques · Spin-orbit Torques in Magnetic…
[25]
[PDF] The cosmological axion domain wall problem - CERN Indico
Oct 23, 2023 · We revisit the domain wall problem for QCD axion models with more than one quark charged under the Peccei-Quinn symmetry.Missing: multiple vacua
[26]
[PDF] Domain walls and Gravitational Waves
Jul 2, 2020 · • Solution to domain wall problem. Gravitional waves from domain ... ⇒ Domain walls would dominate the energy density of the universe.
[27]
Collapsing domain wall networks - IOP Science
Jun 11, 2024 · Abstract: Unstable domain wall (DW) networks in the early universe are cosmologically viable and can emit a large amount of gravitational ...
[28]
[PDF] Domain walls, cosmic strings and monopoles, dark matter
Existence of domain walls would be cosmic catastrophe! These problems are solved by inflation.! Interestingly, infinite domain walls exhibit repulsive ...
[29]
[2307.04710] Remarks on the Axion Domain Wall Problem - arXiv
Jul 10, 2023 · Abstract: Theories in which the Peccei-Quinn phase transition occurs after inflation tend to suffer from problematic domain walls.
[30]
[1909.01604] A Dynamical Solution to the Axion Domain Wall Problem
Sep 4, 2019 · In this paper, we considered a new mechanism which solves those problems by dynamics of multiple scalar fields during/after inflation.
[31]
Gravitational waves from walls bounded by strings in SO(10) model ...
Oct 10, 2023 · Fig. 3 shows the gravitational wave background for the domain wall formation scale varying from 102 to 105 GeV, with G μ = [ 10 − 15 , 10 − 9 ] ...
[32]
[2209.14313] One $μ$ to rule them all: CMB spectral distortions can ...
Sep 28, 2022 · We give analytic estimates for the size of the distortions and outline how to calculate them from first principles. These methods are applied to ...
[33]
The spectrum of gravitational waves from annihilating domain walls
Jul 16, 2025 · We study the production of a cosmological background of gravitational waves (GWs) from such networks, when they annihilate due to a small explicit symmetry ...
[34]
Cosmological Consequences of Domain Walls Biased by Quantum ...
Jan 27, 2025 · We explore the possibility that quantum gravity effects are responsible for violation of the discrete symmetry, triggering the annihilation of the domain wall ...Missing: epsilon | Show results with:epsilon
[35]
A Method for Simulating Chiral Fermions on the Lattice - arXiv
Jun 11, 1992 · B288:342-347,1992 ... Access Paper: View a PDF of the paper titled A Method for Simulating Chiral Fermions on the Lattice, by D.B. Kaplan.Missing: Domain wall
[36]
Residual Chiral Symmetry Breaking in Domain-Wall Fermions - arXiv
Jul 24, 2000 · We find that the induced quark mass is nearly independent of the physical volume, decays exponentially as a function of L_s, and has a ...
[37]
[PDF] 18. LATTICE QUANTUM CHROMODYNAMICS - Particle Data Group
Oct 1, 2016 · Two types of Ginsparg-Wilson fermions are currently being used in large-scale numerical simulations. The first is Domain-wall fermions (DWF).